Why is the derivative of absolute value of x (x*x')/abs(x)

I have no problem that I am trying to solve but simply a question about the derivative of an absolute value equation. I know that the derivative of and absolute value function is (x*x')/(abs(x)) and I understand the process of reaching this equation through the process shown here.

I would like to know why one cannot give the answer of (abs(x))/(x*x') instead of the equation I mentioned earlier. Their graphs are exactly the same and do not have any differences I can find other than the equation.

For example the derivative of abs(x) should be x/abs(x) but the graph of abs(x)/x is defined for all the same values and also returns all the same values and the proper answer. Please help me understand why the latter equation is considered incorrect and not the derivative of the abs(x).

Thanks in advance for your help.1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution1. The problem statement, all variables and given/known data

[edit] What I mean to say is that
$$\frac{x \cdot x'}{|x|} = \frac{|x| \cdot x'}{x}$$
However, I don't think either of these is equal to
$$\frac{|x|}{x \cdot x'}$$
which is the expression you gave above.

Yeah, I tried explaining that to my calc teacher and he just said that it was incorrect because they were not the same. At the time I did not realize I could prove the abs(x)/x to be equal to x/abs(x) so I just let it go.